Indian Statistical Institute, ISI BStat & BMath 2021 UGB Solutions & Discussions
ISI BMath & BSTAT 2021 Subjective Questions UGB: solutions and discussions
Problem 1.
There are three cities each of which has exactly the same number of citizens, say . Every. citizen in each city has exactly a total of
friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).
Topic: The Extremal Principle
Difficulty level: Medium
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 1 UGB Solution
Solution:
From the citizens, take one who has a maximum number
of acquaintances from one of the two other cities. Suppose it is citizen
from the first city, who knows
citizens from the second city. Then
knows
citizens from the third city,
since
. Consider citizen
from the third city, who knows
. If
knows at least one citizen
from the
acquaintances of
in the second city, then
is a triple of mutual acquaintances. But if
knows none of the
acquaintances of
in the second city, then, in this city he does not know more than
citizens, and hence, in the first city, he does not know less than
citizens, which contradicts the choice of
.
Video Solution:
Problem 2.
Let be a function satisfying
. Assume also that
satisfies equations (A) and (B) below.
for all integers .
(i) Determine explicitly the set .
(ii) Assuming that there is a non-zero integer such that
, prove that the set
is infinite.
Topic: Functional Equations
Difficulty Level: Medium
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 2 UGB Solution.
Solution:
Let in (A):
Since we deduce
Let in (B):
Either or
The function takes only two values
and
(both values are really in the range of
, because
).
Assume that there exist a non-zero such that
Then
Take
in (A):
It follows that hence
and
So,
Further, from (A) we find that if
then
By induction, for all
and the set
is infinite.
Video Solution:
Problem 3.
Prove that every positive rational number can be expressed uniquely as a finite sum of the form
where are integers such that
for all
.
Topic: Number Theory
Difficulty Level: Medium
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 3 UGB Solution.
Solution:
Let be a positive rational number. Define
so that
Assume that are defined in such a way that
Define
Since
we necessarily have So, there exists a sequence of integers
such that
and for all
For large enough we have
so
The latter is only possible when
Now assume that there are two representations
with integer coefficients such that
for
Let
be the first index such that
By symmetry we can assume that
Then
Consider inequalities
Hence, the equality is impossible.
Video Solution:
Problem 4.
Let be a differentiable function whose derivative is continuous, and such that
for all
If
is not the identity function, prove that
must be strictly decreasing.
Topic: Calculus
Difficulty Level: Easy
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 4 UGB Solution.
Solution:
We differentiate the relation
for all
hence
preserves the sign on
Assume that
for all
Then
is strictly increasing. For some
we have
(as
is not the identity function). If
then
but this contradicts If
then
but this also contradicts So, the assumption
leads to a contradiction. Then
for all
and
is strictly decreasing.
Video Solution:
Problem 5.
Let and
If for all
, and
then find
Topic: Calculus
Difficulty Level: Medium
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 5 UGB Solution.
Solution:
The relation implies that
All coefficients that correspond to odd indices
are equal to zero. So,
Consider polynomial By assumption,
Observe that is the coefficient near
in
roots of the polynomial
are known:
In particular, and
Then and
Video Solution:
Problem 6.
If a given equilateral triangle of side length
lies in the union of five equilateral triangles of side length
, show that there exist four equilateral triangles of side length
whose union contains
.
Topic: Geometry
Difficulty Level: Medium
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 6 UGB Solution.
Solution:
We show that necessarily Assume that
Consider an equilateral triangle
of side length
The distance between any two points in
is
Now consider an equilateral triangle
of side length
Consider the following configuration:
There are six points in the triangle with pairwise distances Hence, if
then one needs at least six equilateral triangles of side length
to cover an equilateral triangle of side length
So,
and the configuration above shows that
equilateral triangles of side length
are enough to cover an equilateral triangle of side length
Video Solution:
Problem 7.
Let be three real numbers which are roots of a cubic polynomial, and satisfy
and
Suppose
Show that
Topic: Polynomials
Difficulty Level: Easy
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 7 UGB Solution.
Solution 1:
We have
The derivative of
The polynomial is increasing on
decreasing on
increasing on
Then necessarily
Further,
i.e. and
and
Finally,
and
Video Solution:
Problem 8.
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is . The square at the bottom has side length
and the top square has side length
Water is filled in at a rate of
cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?
Topic: Calculus
Difficulty Level:
Indian Statistical Institute, ISI Subjective BStat & BMath 2021 Problem 8 UGB Solution.
Solution :
Denote by the volume of the water after
hours the water started to fill the pond. Then
(cubic meters). Let
be the level of the water and
be the side of the square at the top of the water after
hours the water started to fill the pond.
Extend the truncated pyramid to get the non-truncated pyramid and let be the height of the added part. Then we have the following picture
Similarity of triangles implies relations
Multiplying them we find
Hence,
Now,
Differentiate this:
Then
In two questions I messed up in the last line where we finally write the answer .
Like instead of 19/27 or 19/3(3)^2 in the last question , is wrote 19/3(3) forgot the square , other wise all was fine until the last line …..
And I’m polynomial question , I wrote 1-(9!)^2 instead of plus sign in last again , how much marks would that cost , can I get a 9/10 probably ,so email me
I feel sir made a mistake out there because the constant term of Q(x) is indeed -(9!)^2 as putting 0 in (x-1)(x-4)(x-9)……(x-81)….would give us (-1)(-4)(-9)….(-81) = -(9!)^2
Yes, there were some calculation errors. Thanks for notifying me. Corrected.
But sir the polynomial should be x^2(x^2 – 1^2 )(x^2 – 2^2)… (x^2 – 9^2)+x^2.so the answer should be 1-(9!)^2.
( I have not given the exam this year, I read in class 12and will give isi entrance on 2022 though I have solved this in home and got this expression of the polynomial as result. Please inform me if it is wrong
sir, in the 4th question, since, g(g(x)) = x, g(x) is a function which is its own inverse. there are three such operations which have these property, which is, they cancel their own effect when acted an even number of times, which are as follows:
(i) the identity function : doing nothing to the input.
(ii) g(x) = -x : multiplication by -1 is cancelled by again multiplying it by -1.
and (iii) taking the reciprocal, i.e., g(x) = 1/x or a/x.
since, g(x) is defined from (o, infinity) to (o, infinity) and is not the identity function, it is the reciprocal i.e., g(x) = a/ x , for some positive, a.
so its derivative is -a/x^2 which is always negative as x^2 and a are always positive. so, the function g(x) is strictly decreasing.
is this correct, sir?
But sir the polynomial should be x^2(x^2 – 1^2 )(x^2 – 2^2)… (x^2 – 9^2)+x^2.so the answer should be 1-(9!)^2.
( I have not given the exam this year, I read in class 12and will give isi entrance on 2022 though I have solved this in home and got this expression of the polynomial as result. Please inform me if it is wrong
Sir for problem 5, if we do not write the final answer(-362879) how much mark will be deduced?
In question number 5,what concept did you use and it from what book,i have read enough but this was out of the blue.
thank you in advance
You can’t get everything in a book. You have to come up with some ideas too. Anyways this is basic polynomial stuff.